Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,027 questions
1
vote
1
answer
555
views
The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
- 55
4
votes
1
answer
774
views
SDE-removal of the diffusion coefficients
from math.stackexchange
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have
\begin{align}
dX_t=b(X_t)dt+\sigma dW_t,
\end{align}
...
- 41
2
votes
1
answer
293
views
MMSE estimator expressed through cumulants
I have a linear model $$Y=HX+N,$$ where $H$ is a matrix and $X$ are drawn from $p_X(X)$, and $N$ is Gaussian noise variates.
Now, if $X$ is multivariate Gaussian, then a linear estimator $\hat{X}=A+BY$...
0
votes
1
answer
262
views
Conditional Density of Random Variables
Hi all,
I read recently that for any three continuous random variables, X,Y and Z, the conditional densities are related by the following formula:
$p(x|y) = \int g(x| z) h(z | y ) dz $
where $p(x|...
9
votes
4
answers
1k
views
Symmetries of probability distributions
When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q$...
- 1,655
5
votes
0
answers
202
views
hitting time of a subset
Suppose you have the simple random walk on $L=\mathbb{Z}^n,$ and $A \subset L$ is a subset (in my case, the subset is a union of hyperplanes, so in particular infinite). There is a random variable $T(...
- 96.4k
4
votes
0
answers
274
views
Some constants in Martingale Stein inequality
Dear all,
the following is a special case of Stein inequalities for martingales.
$\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of ...
- 769
6
votes
1
answer
566
views
Area of union of random circles in a plane
If $n$ circles of radius $r$ are drawn at random in the plane such that their centers lie in some region $S$ (we could take $S$ to be a square), what is the expected area $E$ of their union?
Edit: In ...
- 413
2
votes
3
answers
1k
views
The probability that a random number N has at least M factors
That is, how to calculate it given the size of N(that is, logN) and assuming that logN is much greater than M. Its an approximation. There is no exact formula.
I do know that according to the prime ...
- 45
29
votes
5
answers
2k
views
Random walk: police catching the thief
I posted this problem on stackexchange.com,but haven't get a satifactory answer.
This is a problem about the meeting time of several independent random walks on the lattice $\mathbb{Z}^1$:
Suppose ...
- 291
1
vote
2
answers
191
views
Is there a notion of likelihood that incorporates information content?
Consider a random variable $F$ with a distribution parameterized by $\theta$ and another random variable $G$ with a distribution parameterized by a variate of $F$, denoted $f$. Note that $F$ is ...
- 123
2
votes
1
answer
689
views
Expected value of sum of first k out of N weighted Gaussian Random Variables
I am investigating the following problem.
Consider N Normal variables with same mean, but difefrent variances. What is the PDF of the linear combination of the largest K random variables being ...
- 29
0
votes
1
answer
182
views
How to Rigorize an inequalities argument
Context
I'm working on a problem involving Lovasz Local Lemma, for proving that there exists a graph with a certain property.
What I need to prove:
There exists some constant $c$, and functions $p,...
- 3
8
votes
3
answers
1k
views
Expected distance between two points in the plane
Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$...
- 83
10
votes
5
answers
509
views
Path length of ball on tilted, perforated plane
Imagine that an $\epsilon$-radius hole is punched in the plane centered
on every integer-coordinate point.
Now a point "ball" is dropped on the plane at a random spot $p$.
If $p$ has not already ...
- 151k
2
votes
1
answer
235
views
The distance between the centroid of $P$ points and the centroid of a subset of the points
Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points ...
- 55
2
votes
1
answer
151
views
Large Deviations for $\nu_\epsilon = Z_\epsilon\exp\left(-\frac{1}{\epsilon}\Phi(x)\right)d\mu$
Given a probability measure of the form
$$\nu_\epsilon=Z_\epsilon\exp\left(-\frac{1}{\epsilon}\Phi(x)\right)d\mu$$
with $Z$ being the normalizing constant.
Under which conditions on $\mu$ and $\Phi$...
- 1,256
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
2
votes
1
answer
350
views
The equilibrium position for the body of a spider-like spring system after randomly perturbing the anchor positions of its legs
Take $N$ springs, $(s_1, ..., s_N) \in S$ of length $(l_1, ..., l_N)$, and for each spring, label one end "A" and one end "B". Connect the "A" ends of the $N$ springs to a point-like particle on a ...
- 55
0
votes
1
answer
407
views
Transformation of probability space.
Let (\Omega, F, P) be a probability space, which may have atoms (important), S be a set of measure-preserving transformations T:\Omega\to\Omega, that is, such that preimage T^{-1}(A) is measurable ...
- 1
1
vote
1
answer
132
views
Approximating Moment of Sum of RVs
Given
$X_i$ are independent random variables.
$|X_i| < 1$
$E[X_i] = 0$
$X = \sum_i^n X_i$
$var(X)=\sigma$
Prove:
$$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p
Things I've tried:
...
- 11
3
votes
1
answer
599
views
Is positive part of the kernel measurable?
Let $(E,\mathscr E)$ be a measurable space and $Q:E\times \mathscr E\to\Bbb [-1,1]$ be a signed bounded kernel, i.e. $Q_x(\cdot)$ is a finite measure on $(E,\mathscr E)$ for any $x\in E$ and $x\mapsto ...
- 1,655
14
votes
2
answers
2k
views
Markov chains: invariant measures and explosion
The following seems like such an elementary question, but I didn't get anywhere with it.
Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure (...
9
votes
2
answers
2k
views
common dominating measure for a family of measures
Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that
$$\mu_i=f_i \lambda$$
where the $f_i$ are densities (...
- 1,256
3
votes
1
answer
5k
views
With Huffman code, why do we still need Shannon code?
I'm studying information theory by myself.
I'm confused about that since we already have Huffman code, which is the optimal code method, why are Shannon code and some other code still useful?
I ...
- 351
1
vote
0
answers
223
views
Why this two model have same probability distribution?
(1)
Consider the following method of generating a random tree with $n$ nodes.
First expand the root node into two branches.
Then expand one of the two terminal nodes at random.
At time $k$, ...
- 351
3
votes
1
answer
505
views
Large deviations for sums of exponentially distributed random variables.
Take a large integer $R$, and let $(X_j)_{j\geq R}$ be a sequence of exponentially distributed random variables with parameters $\pi_j := j^{1+\alpha}$ ($\alpha>0$), so that $\sum_{j\geq R} \frac{1}...
- 269
7
votes
0
answers
454
views
Does the law of a Feller Process on a non-locally-compact Polish space depend continuously on the initial condition (in Skorohod path-space)?
I am sure this is written down somewhere but cannot find it.
Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a ...
- 888
1
vote
1
answer
1k
views
Exploiting conditional independence working with covariance matrices
I have a Bayesian network where the number of nodes is potentially large. I've conditioned on some of the nodes (observed data) and I'm trying to draw samples from the distribution remaining nodes (...
- 111
2
votes
1
answer
635
views
Azuma's Inequality when the conditions hold with high probability?
In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
- 201
2
votes
1
answer
2k
views
The expected minimum Hamming distance within a set of randomly selected binary strings
If I randomly sample with replacement $P$ times from a set of all possible binary strings of length $L$, what is a good lowerbound on the expected minimum Hamming distance between any two of my $P$ ...
- 67
1
vote
1
answer
148
views
Staggered timing on 2-D random walks by multiple agents
In 2-D lattice random walks by multiple drunks who can't step onto each other, mathematically I would just say the whole cellular automaton updates "at once".
But to simulate this on a computer, I ...
- 507
2
votes
1
answer
479
views
Small geometric progression modulo N
An problem related to integer factorization using the General Number Field Sieve is the following:
Let $N$ be a composite. Must there exist a 5-term geometric progression $\lbrace a_0,a_1,a_2,a_3,...
- 360
12
votes
4
answers
4k
views
Tail bound for Poisson random variable
Is the following fact about Poisson random variables true?
For any $\lambda \in (0,1)$ and integer $k > 0$, if $X$ is a Poisson random variable with mean $k \lambda$, then $\Pr(X < k) \geq e^{...
- 785
1
vote
0
answers
93
views
Potentials of class D
A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...
- 667
3
votes
0
answers
765
views
Bound on a sum involving binomial distribution
Let $f^B_{j,a}(s)$ be the probability mass function of the binomial distribution, that is $f^B_{j,a}(s) = {j \choose s} a^s (1-a)^{j-s}$. And let $F^B_{j+1,b}(s)$ be the cdf of the binomial ...
- 111
4
votes
0
answers
5k
views
E[ | X - Y | ] where X and Y are independent Poisson random variable
What is the expected value of the absolute difference of two independent Poisson variables?
$$E[ |X - Y| ]$$
Seems like an easy question but I haven't found an easy solution.
I've split the double ...
- 41
4
votes
1
answer
352
views
Ising entropy of a finite L_1 x L_2 lattice
We know the entropy per site of the 2-d Ising model from Onsager's solution.
Has anybody also calculated the entropy for a finite rectangle of size L_1 x L_2
with periodic boundary conditions (i.e. on ...
16
votes
3
answers
2k
views
Solving a modified birthday problem at a glance
Modified Birthday Problem: a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal?
Three (...
- 7,831
2
votes
1
answer
646
views
A wrong proof of Squared Bessel process
The squared Bessel process with $\delta$-dimension for $\delta>0$,
denoted by $BESQ^\delta(y)$, is given by
$$d Y_t = \delta t + 2 \sqrt{Y_t} d B_t, \ Y_0 = y\ge 0$$
where $B_t$ is BM under $(\...
- 1,399
5
votes
1
answer
1k
views
Is this process strictly positive?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and $\...
- 1,399
11
votes
1
answer
2k
views
Bounding the entropy of a convolution
Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
user17253
1
vote
1
answer
902
views
Product of densities of a wrapped normal distribution
The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...
- 73
4
votes
0
answers
445
views
Hessians of Fourier transforms of positive radial functions
$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\eW}{\mathscr{W}}$
While investigating the distribution of critical points of random funtions on tori I was lead to ...
9
votes
2
answers
646
views
Rain droplets falling on a table
Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
- 191
8
votes
0
answers
266
views
Fixed marginals of joint distribution: status
One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left (...
- 421
7
votes
4
answers
1k
views
Recent impressive combinatorial developments in probability theory
In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv)
... I suspect that, for at least a decade, the most ...
- 1,302
2
votes
0
answers
113
views
Does this series stopping times marching forward?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider stochastic differential equation
$$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$
Note that, the above SDE has a strong non-negative ...
- 1,399
2
votes
2
answers
944
views
measuring distance between probability measures only at the tail
Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?
Take, for example, the total ...
- 171
9
votes
2
answers
1k
views
Random pseudoprimes vs. primes
(Edit. What I called "pseudoprimes" are known as "Cramér random primes" in the literature,
of which I was unaware.)
Say that a set $S$ of natural numbers is a set of pseudoprimes if they
are (a) ...
- 151k